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P. Lederer, C. Lehrenfeld, J. Schöberl:
"Hybrid Discontinuous Galerkin methods with relaxed H(div)-conformity for incompressible flows";
arXiv.org e-Print archive, 1707 (2017), 02782; 29 S.

We propose a new discretization method for the Stokes equations. The method is an improved version of the method recently presented in [C. Lehrenfeld, J. Sch"oberl, Comp. Meth. Appl. Mech. Eng., 361 (2016)] which is based on an H(div) -conforming finite element space and a Hybrid Discontinuous Galerkin (HDG) formulation of the viscous forces. H(div) -conformity results in favourable properties such as pointwise divergence free solutions and pressure-robustness. However, for the approximation of the velocity with a polynomial degree k it requires unknowns of degree k on every facet of the mesh. In view of the superconvergence property of other HDG methods, where only unknowns of polynomial degree k−1 on the facets are required to obtain an accurate polynomial approximation of order k (possibly after a local post-processing) this is sub-optimal. The key idea in this paper is to slightly relax the H(div) -conformity so that only unknowns of polynomial degree k−1 are involved for normal-continuity. This allows for optimality of the method also in the sense of superconvergent HDG methods. In order not to loose the benefits of H(div) -conformity we introduce a cheap reconstruction operator which restores pressure-robustness and pointwise divergence free solutions and suits well to the finite element space with relaxed H(div) -conformity. We present this new method, carry out a thorough h -version error analysis and demonstrate the performance of the method on numerical examples.